Probe placement optimization in gas turbine engines

ABSTRACT

A method of optimizing probe placement in a turbomachine is disclosed which includes determining wavenumber (Wn) of N dominant wavelets generated by upstream and downstream stators and blade row interactions formed around an annulus, establishing a design matrix A utilized in developing flow properties around the annulus having a dimension of m×(2N+1), iteratively modifying probe positions placed around the annulus and determining a condition number of the design matrix A for each set of probe positions until a predetermined threshold is achieved for the condition number representing optimal probe position, wherein the condition number is defined as norm A·norm A+, wherein A+ represents inverse of A for a square matrix and a Moore-Penrose pseudoinverse of A for a rectangular matrix.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present patent application is related to and claims the prioritybenefit of U.S. Provisional Patent Application Ser. No. 63/073,024,entitled PROBE PLACEMENT OPTIMIZATION IN GAS TURBINE ENGINES, filed Sep.1, 2020, and U.S. Provisional Patent Application Ser. No. 63/073,029,entitled METHOD FOR RECONSTRUCTING NON-UNIFORM CIRCUMFERENTIAL FLOW INGAS TURBINE ENGINES, filed Sep. 1, 2020, the contents of which arehereby incorporated by reference in its entirety into the presentdisclosure.

STATEMENT REGARDING GOVERNMENT FUNDING

None.

TECHNICAL FIELD

The present disclosure generally relates to gas turbine engines and inparticular, to an optimized methodology of probe placement to measurethe mean flow properties such as temperature and pressure.

BACKGROUND

This section introduces aspects that may help facilitate a betterunderstanding of the disclosure. Accordingly, these statements are to beread in this light and are not to be understood as admissions about whatis or is not prior art.

A flow field in a compressor is circumferentially non-uniform. Thecircumferential variations measured in an absolute reference frame areassociated with the wakes from upstream stator row(s), potential fieldsfrom both upstream and downstream stator rows, and their aerodynamicinteractions. In a typical engine or technology development programs,the performance such as thermal efficiency of the engine or component iscommonly characterized using measurements acquired from a few probes atdifferent circumferential locations. However, because the flow in a gasturbine engine is non-uniform along the circumferential direction, thecalculated engine performance using measurements from one probe set canbe different from another probe set with changes in the circumferentiallocations.

Also, stator-stator and rotor-rotor interactions can impact stageperformance. For example, in a 2.5-stage transonic axial compressor a0.1% efficiency variation was seen due to stator-stator interactions anda maximum of 0.7% variation in efficiency was observed caused byrotor-rotor interactions. The effect of stator-stator interactions onstage performance have been investigated using vane clocking, thecircumferential indexing of adjacent vane rows with the same vane count.According to another example, in a 3-stage axial compressor a 0.27-pointvariation in the isentropic efficiency of the embedded stage wasobserved at the design loading condition and a 1.07-point variation inthe embedded stage efficiency was observed at a high loading conditionwith changes in vane clocking configurations. The experimentalcharacterization of stage efficiency is facilitated when similar vanecounts exist because that means that measuring the flow across a singlevane passage will accurately capture the full-annulus performance. Thisis great for research, but it is not a common luxury for realcompressors, in which the stators typically have different vane countsrequiring measurements over several pitches, if not the entire annulus,to accurately capture the circumferential flow variations.

Therefore, to accurately measure temperature and pressure profiles,probes on rakes are typically mounted during operation. To characterizethe performance of a compressor, the rakes are typically equally spacedat several stations (fixed axial positions) around the annulus. At eachstation, the thermodynamic properties acquired from the probes atdifferent locations are averaged to a single value to represent the meanflow property. Historically, a simple area-average has been used becauseof the associated simplicity in implementation. Other averaging methodshave emerged including mass-average, work-average, and momentum-averagemethods during the past few decades, all of which require additionalflow field information. However, regardless of the different averagingmethods used, without the detailed information of flow properties aroundthe full annulus, the accuracy of the averaged value as a representationof the true mean flow property is limited, and understanding how mucherror it can introduce into the calculation of compressor performance isimportant. To answer this question, one prior art conducted a fullannulus URANS simulation in a 3.5-stage axial compressor at midspan andshowed that the circumferentially non-uniform flow can cause more than aone-point error in compressor stage performance measurements. In anotherrecent prior art they investigated the instrumentation errors caused bycircumferential flow variations in an 8-stage axial compressorrepresentative small core compressor of an aero-engine. The analysisshowed that a baseline configuration with 3 equally spaced probes aroundthe annulus yields a maximum of 0.8% error in flow capacity and 2.8points error in compressor isentropic efficiency. Since designers areworking hard to find efficiency improvements on the order of 0.1 points,a 2.8-point uncertainty in efficiency is not sufficient for confirmationof typical performance improvements in technology development programsprobe.

Therefore, it is of great value to resolve the compressor non-uniformcircumferential flow for precise calculation of compressor performance,as well as better prediction of blade forced response. Historically,experimental characterization of the circumferential flow variation isachieved by circumferentially traversing the flow, either utilizing aprobe traverse mechanism or utilizing fixed instrumentation whileactuating the stator rows circumferentially. These approaches involvethe design of complex traverse mechanisms that are challenging to sealand can be costly.

Therefore, there is an unmet need for a novel approach for probeplacement at different circumferential locations to arcuately determineturbine engine performance characteristics such as thermal efficiency ofthe engine.

SUMMARY

A method of optimizing probe placement in a turbomachine is disclosed.The method includes determining wavenumber (Wn) of N dominant waveletsgenerated by upstream and downstream stators and blade row interactionsformed around an annulus. The wavenumber represents how many times awavelet repeats along the annulus. N represents a predetermined numberbased on vane counts of the blade rows. The wavenumber Wn represents thenumber of the complete cycle of a wave over its wavelength along theannulus. The method further includes establishing a design matrix Autilized in developing flow properties around the annulus having adimension of m×(2N+1), where m represents the number of datapoints atdifferent circumferential locations around the annulus. m is greater orequal to 2N+1. Additionally, the method includes iteratively modifyingprobe positions placed around the annulus and determining a conditionnumber of the design matrix A for each set of probe positions until apredetermined threshold is achieved for the condition numberrepresenting optimal probe position, wherein the condition number isdefined as norm A·norm A+, wherein A+represents inverse of A for asquare matrix and a Moore-Penrose pseudoinverse of A for a rectangularmatrix.

BRIEF DESCRIPTION OF DRAWING

FIG. 1 is a schematic of a typical gas turbine engine.

FIG. 2 is a graph of circumferential total pressure field at mid-spanupstream of stator number 6 in a multi-stage axial compressor atdifferent angular positions.

FIG. 3a is a schematic representation of flow interactions stemming fromblades of two rotors and a stator representing blade row interactionswhich result in complex circumferential nonuniform flow patterns.

FIG. 3b is a schematic of a typical multi-stage axial compressor used inhigh-pressure compressor (HPC) assembly of gas turbines.

FIG. 4 is a flowchart depicting steps of defining optimal probepositions based on a minimum condition number according to the presentdisclosure using a Particle swarm optimization algorithm.

FIG. 5 is a graph of a cost function which represent a combination ofcondition number vs. impermissible probe position for differentiterations.

FIG. 6a is a graph similar to FIG. 2 with probe position shown.

FIG. 6b is histogram of condition number vs. wavenumber combinations.

FIG. 7a is a schematic of a flow path of a compressor and distributionof the steady instrumentation.

FIG. 7b is a diagram depicting distribution of the static pressure tapsat a diffuser leading edge.

FIG. 8 is a histogram of condition number vs. wavenumber for 10 selectedwavenumbers.

DETAILED DESCRIPTION

For the purposes of promoting an understanding of the principles of thepresent disclosure, reference will now be made to the embodimentsillustrated in the drawings, and specific language will be used todescribe the same. It will nevertheless be understood that no limitationof the scope of this disclosure is thereby intended.

In the present disclosure, the term “about” can allow for a degree ofvariability in a value or range, for example, within 10%, within 5%, orwithin 1% of a stated value or of a stated limit of a range.

In the present disclosure, the term “substantially” can allow for adegree of variability in a value or range, for example, within 90%,within 95%, or within 99% of a stated value or of a stated limit of arange.

A novel approach is provided in the present disclosure for probeplacement at different circumferential locations to arcuately determineturbine engine performance characteristics such as thermal efficiency ofthe engine. This novel approach aims to reconstruct compressornonuniform circumferential flow field using spatially under-sampled datapoints from a few probes at fixed circumferential locations. Theapproach principally utilizes a Particle Swarm Optimization algorithmfor selection of optimal probe position. Consequently, the methodbridges the gap between sparsely distributed experimental data and thedetailed flow field of a full annulus. Through the two experiments indifferent types of compressors, the method shows great potential inobtaining suitable mean flow properties for performance calculations aswell as resolving the important flow features associated withcircumferential non-uniformity. The method can be disruptive to the gasturbine community concerning: expectations of experimental data; how andwhere to place the probes; and the method to calculate suitable meanflow properties.

A gas turbine engine typically includes three elements including: acompressor, a combustor, and a turbine. Referring to FIG. 1, a schematicof a typical gas turbine engine 100 is shown. The gas turbine engine 100typically includes an intake 102 which includes an air intake 104 havingan initial cross sectional area through which air is allowed into thegas turbine engine 100 at a high rate of speed. The incoming air iscompressed through a compressor 106 which reduces the effective crosssection prior to entering into a combustion zone 108 having one or morecombustion chambers 110. In the one or more combustion chambers 110, thecompressed air is energized and then is directed to turbines 112 priorto being ejected out of an exhaust 114. The compressor 106 or turbines112 include stationary blade rows which are called stators as well asrotating blade rows which are called rotors. Each includes a pluralityof stages. Thus, a stage includes a stator and a rotor. The flow fieldin a compressor or turbine is circumferentially non-uniform due to thewakes from upstream stators, the potential field from both upstream anddownstream stators, and blade row interactions. To demonstrate thisnon-uniformity, reference is made to FIG. 2 which shows thecircumferential total pressure field at mid-span upstream of statornumber 6 in a multi-stage axial compressor. The abscissa represents thecircumferential position along the full annulus and the ordinaterepresents the normalized nondimensional pressure. As described above,this observed non-uniformity is due to the wakes from upstream statorrow(s), potential fields from both upstream and downstream stator rows,and their aerodynamic interactions. These interactions are shown in FIG.3a which provides a schematic representation of flow interactionsstemming from blades of two rotors and a stator representing blade rowinteractions which result in complex circumferential nonuniform flowpatterns.

In theory, the circumferential flow field in turbomachines with aspatial periodicity of 2π can be described in terms of infinite serialwavelets of different wavenumbers:

x(θ)=c ₀+Σ_(i=1) ^(∞)(A _(i) sin(W _(n,i)θ+φ_(i)))   (1)

where x(θ) represents the flow property along the circumferentialdirection,

-   c₀ represents the mean component of x(θ),-   W_(n) is abbreviated for wavenumber,-   W_(n,i) represents the i^(th)wavenumber, and-   A_(i) and φ_(i) represent the magnitude and phase of the wavelet of    the i^(th)wavenumber. A wavenumber is the spatial frequency of a    wave, measured in cycles per unit distance or radians per unit    distance. In the present disclosure, the wave number is also    referred to as the spatial frequency of a wave along the    circumferential direction per 2π.

Furthermore, defining a_(i)=A_(i) cos φ_(i) and b_(i)=A_(i) sin φ_(i),Eqn. (1) can be cast as:

x(θ)=c ₀+Σ_(i=1) ^(∞)(a _(i) sin(W _(n,i)θ)+b _(i) cos(W _(n,i)θ))   (2)

The circumferential flow in a multi-stage compressor is typicallydominated by several wavenumbers. Therefore, instead of using aninfinite number of wavelets described in Eqn. (1), the circumferentialflow in the compressor can be approximated by a few (N) dominantwavelets (where the dominance is measured by the magnitude based on apredetermined threshold weight of magnitude) by the approximation:

x(θ)≈c ₀+Σ_(j=1) ^(N)(a _(j) sin(W _(n,j)θ)+b _(j) cos(W _(n,j)θ))   (3)

The above approximation is an important step toward reconstructing thecircumferential flow field since it reduces the number of unknowncoefficients from infinity in Eqn. (1) to 2N+1 in Eqn. (3).

To solve an equation of 2N+1 unknowns, a minimum of the same amount ofdata points (i.e., measurements from circumferentially positionedprobes) measured at different circumferential locations, θ=(θ₁, θ₂, θ₃,. . . θ_(m)), is required, where θ is the circumferential location alongthe annulus, which can be any value from 0 to 360 in degree (or 0 to 2πin rads). The system can be described with:

AF=x   (4)

where A is known as the design matrix with a dimension of m×(2N+1),

-   F is a vector containing 2N+1 unknown coefficients, and-   x is a m-element vector with all the measurement data points from    different circumferential locations. The mathematical expressions    for A, F, and x are

${A = \begin{pmatrix}{\sin\; W_{n,1}\theta_{1}} & {\cos\; W_{n,1}\theta_{1}} & \ldots & {\sin\; W_{n,N}\theta_{1}} & {\cos\; W_{n,N}\theta_{1}} & 1 \\{\sin\; W_{n,1}\theta_{2}} & {\cos\; W_{n,1}\theta_{2}} & \ldots & {\sin\; W_{n,N}\theta_{2}} & {\cos\; W_{n,N}\theta_{2}} & 1 \\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\{\sin\; W_{n,1}\theta_{m}} & {\cos\; W_{n,1}\theta_{m}} & \ldots & {\sin\; W_{n,N}\theta_{m}} & {\cos\; W_{n,N}\theta_{m}} & 1\end{pmatrix}};$ $\mspace{20mu}{{F = \begin{pmatrix}a_{1} \\b_{1} \\\vdots \\a_{N} \\b_{N} \\c_{0}\end{pmatrix}};}$ $\mspace{20mu}{x = \begin{pmatrix}{x\left( \theta_{1} \right)} \\{x\left( \theta_{2} \right)} \\\vdots \\{x\left( \theta_{m} \right)}\end{pmatrix}}$

To solve for the N wavenumbers of interest described in Eqn. (4), thenumber of the data points in vector x must be equal or greater than thenumber of unknown coefficients, or m≥2N+1. However, in practice, thereconstructed signal contains errors due to the uncertainties in x(θ),and it is important to evaluate the confidence in the reconstructedsignal, which requires additional data points in x(θ). Therefore, aminimum of 2N+2 measurement points is recommended to characterize Nwavenumbers of interest. Therefore, the wavenumber of interest is firstdetermined and then probe placement optimization is carried out with thedetermined wavenumber. The probe placement optimization is carried outusing a Particle Swarm Optimization algorithm. Each of these steps arediscussed below in greater detail.

Determining wavenumber: Even though the circumferential flow incompressors can be approximated using a few dominant wavelets, resolvingall of these wavenumbers can still be challenging. In practice, due tothe cost and blockage associated with each probe, there is usually alimit on the number of probes allowed per blade row. Typically, a rangeof 3 to 8 rakes/probes per blade row is achievable. However, accordingto Eqn. (3), a set of 4, 6, and 8 probes can resolve 1, 2, and 3wavenumbers, respectively. Thus, an intelligent selection of the mostimportant wavenumbers is needed to assure the best results forreconstructing the signal from a limited number of probes. The mostimportant wavenumbers can be determined with the help of informationfrom either reduced-order modeling or high-fidelity computational fluiddynamics simulations. For cases with no information available except forairfoil counts, recommended guidelines based on previous research ofmulti-stage interactions for representative wavenumber selection are:

1. Upstream and downstream vane counts;

2. Differences of the upstream and downstream vane counts;

3. Wavenumbers associated with low-count stationary component (i.e.upstream and downstream struts for the front and rear stages).

-   For instance, FIG. 3b is a schematic of a typical multi-stage axial    compressor used in high-pres sure compressor (HPC) assembly of gas    turbines. The dominant wavenumbers for the flow downstream of stator    2 include: 1) 38 (stator 1 vane count), 44 (stator 2 vane count),    and 50 (stator 3 vane count); 2) 8 (the difference of upstream vane    count) and 6 (the difference of the downstream vane count); and 3) 4    (the vane count of the upstream struts), which is not shown in the    figure. Therefore, a selection of 6 wavelets is appropriate. The    process of selecting wavelet and wavenumber is further demonstrated    in Table 1, below.

TABLE 1 Selection of wavelet and wavenumber No. of Wavelets WavenumberConsideration Criterion 1 38 Upstream blade row (s1) 1st vane count 2 44Vane count of itself (S2) 1st 3 50 Downstream blade row (S3) 1st vanecount 4 8 S2-S1 = 44-38 = 8 2nd 5 6 S3-S2 = 50-44 = 6 2nd 6 4 Vane countof inlet Struts 3rdAs a result, 6 dominant wavelets can be selected along with wavenumbersare 38, 44, 50, 8, 6, 4, respectively.

With selection of the wavenumbers of interest, a condition number of thedesign matrix describes how well the probes are distributed to capturethe wavenumbers of interest and determine the confidence interval of thereconstructed flow field, making it an important parameter for theselection of probe locations.

In the field of numerical analysis, the condition number of a functionmeasures how much the output value of the function can change for asmall change in the input argument. This parameter is used to measurehow sensitive a function is to changes or errors in the input, and howmuch error in the output results from an error in the input. A systemwith a low condition number is said to be well-conditioned, while asystem with a high condition number is said to be ill-conditioned. Inthe present disclosure, the condition number gauges how sensitive thereconstructed flow is to the errors in the probe placement. Theobjective of the probe placement optimization is to minimize thecondition number of the design matrix.

The condition number of a matrix is calculated using the formula:

k=∥A∥∥A ⁺∥   (5)

where A⁺ is the inverse of matrix A for a square matrix and theMoore-Penrose pseudoinverse of matrix A for a rectangular matrix. Thedouble-line represents the norm of a matrix. Simply put, the norm of amatrix represents the strength (i.e., the value of the matrix). Tocalculate a norm of a matrix four conditions must be met: 1) the norm isequal or greater than 0 (the norm can be 0 only and only if the matrixis a 0 matrix); 2) scalar property (i.e., ∥kA∥=|k|·∥A∥); 3) additiveproperty (i.e., ∥A+B∥≤∥A∥+∥B∥; and 4) multiplicative property (i.e.,∥A·B∥∥A∥·∥B∥). There are several matrix norms that can be used to definethe condition number in equation 5. For example, one norm is the maximumsum of absolute numbers in each column (i.e., absolute values in eachcolumn are added together to generate one or more sums, and the maximumis the one norm). The infinity norm is the maximum sum of absolutenumber in each row (i.e., absolute values in each row are added togetherto generate one or more sums, and the maximum is the infinity norm).Euclidean norm is the square root of the sum of all squares (i.e., eachentry in the matrix is squared and added together, the Euclidean norm isthe square root of that sum).

The two norm is a bit more complicated. The two norm is the square rootof the maximum eigenvalue of the matrix (A^(T)·A), where A^(T) is thetranspose of A for a real matrix. For a complex matrix, A^(T) isreplaced with the Hermitian conjugate of A. Eigenvalues of a matrix isdefined by solving det(A^(T)·A−δI)=0 where, det represents thedeterminant of the entity in the parentheses, 6 represents theeigenvalues, and I represents the identity matrix. Thus, in order tofind the two norm of matrix A, first the eigenvalues of matrix A^(T)·Ais found, then the maximum eigenvalue is chosen, and then the squareroot of that maximum eigenvalue represents the two norm of the matrix A.There are still yet other norms including max norm (which is the largestabsolute value of all the entries in a matrix), P norm, and other normsknown to a person having ordinary skill in the art. Any of these normscan be used to calculate the condition number of the matrix A accordingto equation 5.

In the present disclosure, the condition number of design matrix A isdetermined by both probe location, θ, and the wavenumbers of interest,W_(n). The value of the condition number of the design matrix can varyfrom one to infinity. As discussed above, a system with a largecondition number can result in excessive error in the reconstructedsignal.

Applying the Particle Swarm Optimization algorithm: Particle swarmoptimization (PSO) is a known optimization technique for solving globaloptimization problems due to its high efficiency of convergence. It wasfirst introduced for simulation of simplified animal social behaviorssuch as bird flocking. In the PSO algorithm, a potential solution iscalled a particle, which has two representative parameters including theposition and velocity. The optimization starts with an initialpopulation of particles and then moves these particles around in thesearch-space. The movement of each particle is influenced by its localbest-known position as well as the global best-known position in theentire search space. As a result, the swarm is iteratively moving towardthe best solution.

In the present disclosure, PSO is used to search for the optimal probepositions that yields the smallest condition number of the designmatrix. Therefore, the design variables (parameters being optimized) arethe circumferential positions of probes, θ, and the objective functionis described using:

f _(obj) =k(θ, W _(n))+f _(constraint)   (6)

where f constraint represents the value of the constraint function fromconsiderations of geometric constraints for placing probes. Tworepresentative constraints in turbomachines include minimum spacingbetween adjacent probes and restricted areas due to casing fixtures.Probes in turbomachines are typically casing-mounted through a varietyof instrumentation ports. A minimum probe spacing is, therefore,necessary for practical implementation. The formula for minimum probespacing is described:

|Δθ_(j,i)|=|θ_(j)−θ_(i)|≥θ_(min)   (7)

where θ_(i) and θ_(j) represent the circumferential position of thei^(th) and j^(th) probes, respectively,

-   Δθ_(j,i) is spacing between the i^(th) and j^(th) probes, and-   θ_(min) represents the minimum probe spacing allowed.

Additionally, in many scenarios, it may not be possible to installprobes at all positions around the circumference due to fixtures orobstructions on certain regions of the casing. A constraint is,therefore, required to prevent probes from being placed in thesecircumferential ranges. The formula for constraints due to casingfixtures is described:

θ_(i){tilde over (∈)}[θ*_(1,min), θ*_(1,max)]|[θ*_(2,min), θ*_(2,max)] .. . |[θ*_(p,min), θ*_(p,max)]   (8)

in which, θ*_(p,min) and θ*_(p,max) represents the minimum and maximumfixture location for the p^(th) fixture. During the optimizationprocess, if the position of any probe violates any of the constraints, a“penalty” or “cost” will be assigned to the constraint function toprevent probe placement in that region. At last, it is worth noting thatthe PSO used according to the present disclosure can also be exchangedby other global optimization techniques for probe optimization.

Referring to FIG. 4, a flowchart 400 is shown depicting the steps usingthe PSO algorithm for optimization of probe placement. Afterinitialization 402, the method of the present disclosure receives awavenumber of interest and any constraint areas as discussed above andas represented by block 404. The method then generates an initial probeplacement according to a predetermined positioning pattern (which is notoptimized), as represented by block 406. This pattern may be randomlychosen, or based on some apriori knowledge, e.g., a probe at each stage.An algorithm as indicated by block 408 is then applied. The PSOalgorithm as indicated by block 408 is shown in the inset and is furtherdescribed below. The output of the optimization algorithm as indicatedby block 408 includes optimal positions of the probes as indicated bythe block 412. Based on the probe position output of the optimizationalgorithm, the condition number is calculated. If the condition numbersatisfies a predetermined threshold as indicated by the block 414, theprobe positions constitute the optimum probe placement as indicated bythe block 416 and the probe placement algorithm ends as indicated by theblock 418. If, however, the calculated condition number does not satisfythe predetermined threshold as test in block 414, then a new generation(iteration) of probe placement is generated based on the particlevelocity of the last iteration (shown with a dashed line). Objectivecriteria, according to one embodiment of the present disclosure, for thecondition number are:

-   k≤2.0 if No. of Wn<3;-   k≤4.0 if 3≤No. of Wn<5; and-   k≤6.0 if No. of Wn 5.

The algorithm shown in the inset of FIG. 4 is known by a person havingordinary skill in the art. Several state-of-the-art optimizationalgorithms such as Particle Swarm Optimization (PSO), simulatedannealing (SA) algorithms, Genetic algorithms (GAs), evolutionaryalgorithms (EAs) etc., can be used to find the optimal probe positions.

To demonstrate the efficacy of this novel approach, an actual reductionto practice was carried out using the particle swarm optimization (PSO)algorithm, in which a particle swarm size of 5,000 was chosen, and theoptimization was run for 100 iterations. Referring to FIG. 5, a graph ofa cost function vs. iteration number is provided which shows the changein the value of objective function during one optimization run. Resultsshow that the value of the objective function decreases quickly duringthe first 20 iterations and gradually settles around 19 after 60iterations. Based on this observation, a selection of 100 iterations isa proper number, according to one embodiment of the present disclosure.

The final probe positions from the run shown in FIG. 5 is indicated bythe circles in FIG. 6a . The probe positions are labeled as P1, P2, P3 .. . respectively. One evident feature associated with the optimizedprobe set is that they are not equally spaced. The maximum probe spacingfalls between P3 and P4, with a value of 62°, while the minimum probespacing is 20°. This non-uniform probe spacing allows forcharacterization of all wavenumbers of interest. In addition, FIG. 6bshows the values of condition number for all combinations of wavenumbersof interest. The abscissa indicates the combinations of all wavenumbersof interest while the ordinate represents the condition number. As shownin FIG. 6b , the condition numbers are fairly constant for allcombinations of wavenumbers. This is expected and also consistent withthe nature of the objective function, which had a constant weightingfactor for all combinations of wavenumbers. The largest condition numberfor all the combinations of wavenumbers is less than 2. This indicatesthat the optimized probe set can discern all the wavenumbers ofinterest. Thus, the PSO algorithm is capable of optimizing probepositions effectively, yielding small condition numbers for allwavenumbers of interest.

To further provide a proof of this concept, a compressor was used toprovide an actual reduction to practice examples. The objective is todetermine optimal sensor placement. The flow path of the compressor anddistribution of the steady instrumentation is shown in FIG. 7a . Theentire stage includes an inlet housing, a transonic impeller, a vaneddiffuser, a bend, and de-swirl vanes. The inlet housing delivers theflow to the impeller eye. The impeller is backswept and has 17 mainblades plus 17 splitters. The diffuser includes 25 aerodynamicallyprofiled vanes. The compressor design speed is about 45,000 rpm, and theentire stage produces a total pressure ratio near 6.5 at designcondition. Steady performance of the compressor stage is characterizedusing the total pressure and total temperature measurements atcompressor inlet (station 1) and de-swirl exit (station 5), and staticpressure taps are located throughout the flow passage to characterizethe stage and component static pressure characteristics, as shown inFIG. 7 a.

The distribution of the static pressure taps at the diffuser leadingedge is shown in FIG. 7b . There are a total of nine static pressuretaps placed non-uniformly along the circumferential direction. Each ofthem is placed in a different diffuser passage at a different pitchwiselocation from 10% to 90% pitch. Details of the circumferential andpitchwise locations for these pressure taps are shown in Table 2.

TABLE 2 DIFFUSER LEADING EDGE STATIC PRESSURE TAP LOCATIONSCircumferential Pitchwise Description Position (deg) Position (%)Passage No. P1 52.0 60 4 P2 85.1 90 6 P3 103.9 20 8 P4 165.8 50 12 P5198.9 80 14 P6 217.6 10 16 P7 279.5 40 20 P8 312.7 70 22 P9 350.1 30 25

Based on the above teachings for selection of wavenumbers of mostimportance, a total of ten wavenumbers of interest were selected. Theseinclude the first two harmonics from the wakes at station 1 caused bythe struts and rakes (Wn=4 and 8), the first five harmonics of thediffuser counts (Wn=25, 50, 75, 100, and 125), and the interactionsbetween the compressor inlet struts and the vaned diffuser (Wn=21, 17,and 34). The condition numbers of the probe set for the 10 selectedwavenumbers are shown in FIG. 8. The values of all the condition numbersfall in the range between 1.0 to 2.0 indicating the probe set is able tocharacterize all wavenumbers of interest. However, it is worth notingthat this is a unique case. For instance, out of the multiple probe setsinstrumented along the flow path at different stations (impeller exit,diffuser leading edge, etc.), only the probe set located at the diffuserleading edge yields a reasonable condition number.

Those having ordinary skill in the art will recognize that numerousmodifications can be made to the specific implementations describedabove. The implementations should not be limited to the particularlimitations described. Other implementations may be possible.

1. A method of optimizing probe placement in a turbomachine, comprising:determining wavenumber (Wn) of N dominant wavelets generated by upstreamand downstream stators and blade row interactions formed around anannulus, wherein the wavenumber represents how many times a waveletrepeats along the annulus, wherein N is a predetermined number based onvane counts of the blade rows and the wavenumber Wn represents thenumber of the complete cycle of a wave over its wavelength along theannulus; establishing a design matrix A utilized in developing flowproperties around the annulus having a dimension of m×(2N+1), where mrepresents the number of datapoints at different circumferentiallocations around the annulus, wherein m is greater or equal to 2N+1,iteratively modifying probe positions placed around the annulus anddetermining a condition number of the design matrix A for each set ofprobe positions until a predetermined threshold is achieved for thecondition number representing optimal probe position, wherein thecondition number is defined as norm A·norm A+, wherein A+ representsinverse of A for a square matrix and a Moore-Penrose pseudoinverse of Afor a rectangular matrix.
 2. The method of claim 1, wherein thecondition number represents changes in a circumferential flow propertyfor a predetermined change in probe positions.
 3. The method of claim 2,wherein the predetermined change in probe positions is defined asbetween ±θ_(c) where θ_(c) is between about 0 and 360 in degree.
 4. Themethod of claim 3, wherein the condition number is less than or equal toabout 2.0 for wavenumbers less than
 3. 5. The method of claim 3, whereinthe condition number is less than or equal to about 4.0 for wavenumbersbetween 3 and
 5. 6. The method of claim 3, wherein the condition numberis less than or equal to about 6.0 for wavenumbers greater than
 5. 7.The method of claim 1, where A is determined based on AF=x , wherein Fis a vector containing 2N+1 unknown coefficients, and x is an m-elementvector representing a desired flow property around the annulus.
 8. Themethod of claim 2, wherein ${A = \begin{pmatrix}{\sin\; W_{n,1}\theta_{1}} & {\cos\; W_{n,1}\theta_{1}} & \ldots & {\sin\; W_{n,N}\theta_{1}} & {\cos\; W_{n,N}\theta_{1}} & 1 \\{\sin\; W_{n,1}\theta_{2}} & {\cos\; W_{n,1}\theta_{2}} & \ldots & {\sin\; W_{n,N}\theta_{2}} & {\cos\; W_{n,N}\theta_{2}} & 1 \\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\{\sin\; W_{n,1}\theta_{m}} & {\cos\; W_{n,1}\theta_{m}} & \ldots & {\sin\; W_{n,N}\theta_{m}} & {\cos\; W_{n,N}\theta_{m}} & 1\end{pmatrix}};$ $\mspace{20mu}{{F = \begin{pmatrix}a_{1} \\b_{1} \\\vdots \\a_{N} \\b_{N} \\c_{0}\end{pmatrix}};}$ $\mspace{20mu}{{x = \begin{pmatrix}{x\left( \theta_{1} \right)} \\{x\left( \theta_{2} \right)} \\\vdots \\{x\left( \theta_{m} \right)}\end{pmatrix}},}$ where θ_(i) represents probe location.
 9. The methodof claim 1, wherein the condition number of A is based on one norm whichrepresents a maximum sum of absolute numbers in each column of A. 10.The method of claim 1, wherein the condition number of A is based oninfinity norm which represents a maximum sum of absolute number in eachrow of A.
 11. The method of claim 1, wherein the condition number of Ais based on Euclidean norm which represents a square root of a sum ofall squares of each entry in A.
 12. The method of claim 1, wherein thecondition number of A is based on two norm which represents a squareroot of a maximum eigenvalue of the matrix (A^(T)·A), where A^(T) is atranspose of A for a real matrix and a Hermitian conjugate of A for acomplex A, wherein eigenvalues of a matrix is defined by solvingdet(A^(T)·A−δI)=0 where, det represents a determinant of a matrix, δrepresents the eigenvalues, and I represents the identity matrix. 13.The method of claim 1, wherein the condition number of A is based on maxnorm which represents the largest absolute value of all the entries inA.
 14. The method of claim 1, wherein the iteratively modifying theprobe positions is selected from the group consisting of a particleswarm optimization algorithm, a simulated annealing algorithm, a geneticalgorithm, and an evolutionary algorithm.
 15. A method of evaluationplacement of probes in a turbomachine, comprising: receiving a pluralityof probe positions each associated with a probe; determining wavenumber(Wn) of N dominant wavelets generated by upstream and downstream statorsand blade row interactions formed around an annulus, wherein thewavenumber represents how many times a wavelet repeats along theannulus, wherein N is a predetermined number based on vane counts of theblade rows and the wavenumber Wn represents the number of the completecycle of a wave over its wavelength along the annulus; establishing adesign matrix A utilized in developing flow properties around theannulus having a dimension of m×(2N+1), where m represents the number ofdatapoints at different circumferential locations around the annulus,wherein m is greater or equal to 2N+1, determining a condition number ofthe design matrix A for the plurality of probe positions, wherein thecondition number is defined as norm A·norm A+, wherein A+ representsinverse of A for a square matrix and a Moore-Penrose pseudoinverse of Afor a rectangular matrix.
 16. The method of claim 15, where A isdetermined based on AF=x, wherein F is a vector containing 2N+1 unknowncoefficients, and x is an m-element vector representing a desired flowproperty around the annulus.
 17. The method of claim 2, wherein${A = \begin{pmatrix}{\sin\; W_{n,1}\theta_{1}} & {\cos\; W_{n,1}\theta_{1}} & \ldots & {\sin\; W_{n,N}\theta_{1}} & {\cos\; W_{n,N}\theta_{1}} & 1 \\{\sin\; W_{n,1}\theta_{2}} & {\cos\; W_{n,1}\theta_{2}} & \ldots & {\sin\; W_{n,N}\theta_{2}} & {\cos\; W_{n,N}\theta_{2}} & 1 \\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\{\sin\; W_{n,1}\theta_{m}} & {\cos\; W_{n,1}\theta_{m}} & \ldots & {\sin\; W_{n,N}\theta_{m}} & {\cos\; W_{n,N}\theta_{m}} & 1\end{pmatrix}};$ $\mspace{20mu}{{F = \begin{pmatrix}a_{1} \\b_{1} \\\vdots \\a_{N} \\b_{N} \\c_{0}\end{pmatrix}};}$ $\mspace{20mu}{{x = \begin{pmatrix}{x\left( \theta_{1} \right)} \\{x\left( \theta_{2} \right)} \\\vdots \\{x\left( \theta_{m} \right)}\end{pmatrix}},}$ where θ_(i) represents probe location.
 18. The methodof claim 15, wherein the condition number of A is based on one normwhich represents a maximum sum of absolute numbers in each column of A.19. The method of claim 15, wherein the condition number of A is basedon infinity norm which represents a maximum sum of absolute number ineach row of A.
 20. The method of claim 15, wherein the condition numberof A is based on two norm which represents a square root of a maximumeigenvalue of the matrix (A^(T)·A), where A^(T) is a transpose of A fora real matrix and a Hermitian conjugate of A for a complex A, whereineigenvalues of a matrix is defined by solving det(A^(T)·A−δI)=0 where,det represents a determinant of a matrix, δ represents the eigenvalues,and I represents the identity matrix.